{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "wl0O-vD3RwdT",
        "outputId": "13990f19-8af7-4f28-cb69-743576ea69bf"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Collecting openai\n",
            "  Downloading openai-0.27.8-py3-none-any.whl (73 kB)\n",
            "\u001b[?25l     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m0.0/73.6 kB\u001b[0m \u001b[31m?\u001b[0m eta \u001b[36m-:--:--\u001b[0m\r\u001b[2K     \u001b[90m━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━\u001b[0m \u001b[32m73.6/73.6 kB\u001b[0m \u001b[31m6.7 MB/s\u001b[0m eta \u001b[36m0:00:00\u001b[0m\n",
            "\u001b[?25hCollecting backoff\n",
            "  Downloading backoff-2.2.1-py3-none-any.whl (15 kB)\n",
            "Requirement already satisfied: requests>=2.20 in /usr/local/lib/python3.10/dist-packages (from openai) (2.27.1)\n",
            "Requirement already satisfied: tqdm in /usr/local/lib/python3.10/dist-packages (from openai) (4.65.0)\n",
            "Requirement already satisfied: aiohttp in /usr/local/lib/python3.10/dist-packages (from openai) (3.8.4)\n",
            "Requirement already satisfied: urllib3<1.27,>=1.21.1 in /usr/local/lib/python3.10/dist-packages (from requests>=2.20->openai) (1.26.16)\n",
            "Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.10/dist-packages (from requests>=2.20->openai) (2023.5.7)\n",
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            "Requirement already satisfied: attrs>=17.3.0 in /usr/local/lib/python3.10/dist-packages (from aiohttp->openai) (23.1.0)\n",
            "Requirement already satisfied: multidict<7.0,>=4.5 in /usr/local/lib/python3.10/dist-packages (from aiohttp->openai) (6.0.4)\n",
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            "Requirement already satisfied: aiosignal>=1.1.2 in /usr/local/lib/python3.10/dist-packages (from aiohttp->openai) (1.3.1)\n",
            "Installing collected packages: backoff, openai\n",
            "Successfully installed backoff-2.2.1 openai-0.27.8\n"
          ]
        }
      ],
      "source": [
        "!pip install openai backoff\n",
        "import argparse\n",
        "import numpy as np\n",
        "import torch.nn.functional as F\n",
        "from pathlib import Path\n",
        "from tqdm import tqdm\n",
        "import random\n",
        "import json\n",
        "import os\n",
        "import openai\n",
        "import backoff\n",
        "from tenacity import retry, stop_after_attempt, wait_chain, wait_fixed\n",
        "\n",
        "\n",
        "completion_tokens = prompt_tokens = 0\n",
        "\n",
        "openai.api_type = \"azure\"\n",
        "openai.api_base = \"your_api_base\"\n",
        "openai.api_key = \"your_api_key\"\n",
        "\n",
        "api_key = openai.api_key\n",
        "\n",
        "@backoff.on_exception(backoff.expo, openai.error.OpenAIError)\n",
        "def chatcompletions_with_backoff(**kwargs):\n",
        "    # print(f\"params: {kwargs}\")\n",
        "    return openai.ChatCompletion.create(**kwargs)\n",
        "\n",
        "@backoff.on_exception(backoff.expo, openai.error.OpenAIError)\n",
        "def completions_with_backoff(**kwargs):\n",
        "    # print(f\"params: {kwargs}\")\n",
        "    return openai.Completion.create(**kwargs)\n",
        "\n",
        "\n",
        "@retry(wait=wait_chain(*[wait_fixed(3) for i in range(3)] +\n",
        "                       [wait_fixed(5) for i in range(2)] +\n",
        "                       [wait_fixed(10)]))\n",
        "def gptchatcompletion_with_backoff(**kwargs):\n",
        "    return openai.ChatCompletion.create(**kwargs)\n",
        "\n",
        "\n",
        "def gpt4_with_history(messages, model=\"gpt-4\", temperature=0.7, max_tokens=800, n=1, stop=None) -> list:\n",
        "    outputs = []\n",
        "    openai.api_version = \"2023-03-15-preview\"\n",
        "    while n > 0:\n",
        "        cnt = min(n, 20)\n",
        "        n -= cnt\n",
        "        res = chatcompletions_with_backoff(engine=model, messages=messages, temperature=temperature, max_tokens=max_tokens, n=cnt, stop=stop)\n",
        "        outputs.extend([choice[\"message\"][\"content\"] for choice in res[\"choices\"]])\n",
        "    return outputs, messages\n",
        "\n",
        "\n",
        "def set_seed(seed):\n",
        "    random.seed(seed)\n",
        "    np.random.seed(seed)\n",
        "\n",
        "set_seed(42)"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 7\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we construct some extremely difficult CSPs (Constraint Satisfaction Problems), how can we prove that they cannot be solved within polynomial time? Please provide some possible ideas.'''},\n",
        "]\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "PNdawSQLqn1Z",
        "outputId": "1e3634e6-fa03-43b9-ffd1-9c55751757bd"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"Proving that certain Constraint Satisfaction Problems (CSPs) cannot be solved within polynomial time typically involves showing that the problem in question is NP-hard or NP-complete. Here are some possible ideas on how to prove that an extremely difficult CSP cannot be solved within polynomial time:\\n\\n1. Reduction: One common approach is to show that an existing known NP-hard or NP-complete problem can be reduced to the given CSP in polynomial time. This would imply that if the CSP could be solved in polynomial time, then the known NP-hard or NP-complete problem could also be solved in polynomial time, which contradicts the current understanding of computational complexity.\\n\\n2. Gadgets and Compositions: Another technique involves constructing gadgets or smaller components that represent parts of known NP-hard or NP-complete problems. By combining these gadgets, you can create instances of the given CSP that essentially replicate the harder problem. This way, if there was a polynomial-time solution to the CSP, it would also solve the harder problem, which is a contradiction.\\n\\n3. Phase Transitions: In some cases, it might be possible to identify phase transitions in the problem instances, where the difficulty of solving the problem rapidly changes from easy to hard as some parameter varies. By studying these transitions, you might be able to show that certain instances of the CSP are inherently difficult and cannot be solved in polynomial time.\\n\\n4. Lower Bound Proofs: It might also be possible to prove lower bounds on the time complexity of the CSP under certain computational models, such as decision trees or algebraic circuits. If you can show that the problem requires a superpolynomial number of operations in the chosen model, this would provide evidence that the problem is unlikely to be solvable in polynomial time.\\n\\n5. Average-Case Complexity: Proving that a problem is hard in the worst-case might not be enough to establish its intractability in practice. In this case, you could consider analyzing the average-case complexity of the CSP. If you can show that even on average, the problem instances require superpolynomial time to solve, this would provide stronger evidence of the CSP's inherent difficulty.\\n\\n6. Parameterized Complexity: Another approach is to study the parameterized complexity of the CSP, which involves analyzing the problem complexity as a function of certain problem-specific parameters. If you can show that the problem remains hard even when these parameters are small, it would provide further evidence of the problem's intractability.\\n\\nIt's important to note that proving a problem is not solvable in polynomial time is often difficult and sometimes an open question in computer science. However, the above ideas can be a starting point for establishing the computational complexity of extremely difficult CSPs.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 8\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we construct some extremely difficult CSPs (Constraint Satisfaction Problems), how can we prove that they cannot be solved within polynomial time? Please provide some possible ideas.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Phase Transitions: In some cases, it might be possible to identify phase transitions in the problem instances, where the difficulty of solving the problem rapidly changes from easy to hard as some parameter varies. By studying these transitions, you might be able to show that certain instances of the CSP are inherently difficult and cannot be solved in polynomial time.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! How to prove that some CSPs cannot be solved in polynomial time by studying phase transition?'''},\n",
        "]\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "N97RD91RKPLV",
        "outputId": "7ecab689-d933-43e9-ef6c-5f79fa291099"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"To prove that some CSPs cannot be solved in polynomial time by studying phase transitions, you can follow these steps:\\n\\n1. Identify a parameter that affects the problem's difficulty: The first step is to identify a parameter associated with the CSP that influences its difficulty. This parameter could be related to the problem's size, density, or some other structural property. For example, in graph coloring problems, the ratio of edges to nodes may be a relevant parameter.\\n\\n2. Analyze the phase transition: Investigate how the problem's difficulty changes as the parameter varies. Observe at which point the problem transitions from being easy to solve to becoming hard. This is the phase transition point. Typically, phase transitions in CSPs exhibit a sudden change in the solvability of problem instances.\\n\\n3. Show the existence of hard instances: Prove that there exist instances of the CSP near the phase transition point that are difficult to solve. You can do this by showing that the problem instances exhibit certain properties or structures that make them inherently hard. This might involve relating the CSP to a well-known hard problem, such as an NP-complete problem, or establishing the presence of a lower bound on the complexity of the instances.\\n\\n4. Relate the difficulty to polynomial time solvability: With the existence of hard instances established, you must now relate this difficulty to the inability to solve the problem in polynomial time. One way to achieve this is by showing that any algorithm capable of solving the hard instances in polynomial time would also be able to solve other known hard problems in polynomial time, which is unlikely or considered impossible under the widely believed conjecture P ≠ NP.\\n\\nKeep in mind that this method does not provide a formal proof that the problem is not solvable in polynomial time, but it can provide strong evidence to support the intractability of the CSP under consideration. To obtain a more rigorous proof, you would need to establish a direct connection between your CSP and a known hard problem, such as by showing a reduction from an NP-complete problem to your CSP.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 9\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we prove that there exist instances of the CSP near the phase transition, can we prove that some of them cannot be solved within polynomial time? If not, how to construct more complex instances near the phase transition?'''},\n",
        "]\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8M--3LxMaC-Y",
        "outputId": "44cc3e68-1300-4e16-bac6-01f3d64d3fb7"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "['The phase transition in the context of Constraint Satisfaction Problems (CSPs) refers to the phenomenon where the probability of a problem being solvable changes abruptly as a certain parameter is varied. For many CSPs, this parameter is the constraint density (ratio of constraints to variables). The instances of CSPs near the phase transition are generally considered to be the hardest to solve. However, proving that some of these instances cannot be solved within polynomial time is a challenging task.\\n\\nTo date, there is no general proof that instances of CSPs near the phase transition cannot be solved within polynomial time. The main reason is the lack of a definitive proof for the P vs NP problem, which asks whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. Since many CSPs are NP-complete, proving that some instances cannot be solved within polynomial time would imply that P ≠ NP, which remains an open question.\\n\\nRegarding the construction of more complex instances near the phase transition, one can use the following strategies:\\n\\n1. Generate instances with a higher number of variables and constraints: Increasing the size of the problem will generally make it more difficult to solve. However, this might also push the instance further away from the phase transition.\\n\\n2. Introduce hidden structure: Some instances of CSPs may appear to be random but contain hidden structures that make them difficult to solve. For example, one can generate a problem that consists of multiple smaller, interconnected CSPs that are hard to solve independently. The difficulty of solving the overall problem will be compounded by the difficulty of solving each subproblem.\\n\\n3. Vary the type of constraints: Combining different types of constraints can lead to more complex instances. For example, one can mix algebraic, geometric, and logical constraints in a single problem.\\n\\n4. Exploit symmetries and other properties: Some instances of CSPs can be made more difficult by exploiting symmetries or other properties that make them harder to solve. For example, problems with a high degree of symmetry may be difficult for certain algorithms to handle efficiently.\\n\\nIn summary, while proving that instances of CSPs near the phase transition cannot be solved within polynomial time is an open problem, there are several strategies for constructing more complex instances near the phase transition. These strategies involve increasing the size of the problem, introducing hidden structures, combining different constraint types, and exploiting symmetries or other properties.']\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 10\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we prove that there exist instances of the CSP near the phase transition, can we prove that some of them cannot be solved within polynomial time? If not, how to construct more complex instances near the phase transition?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Exploit symmetries and other properties: Some instances of CSPs can be made more difficult by exploiting symmetries or other properties that make them harder to solve. For example, problems with a high degree of symmetry may be difficult for certain algorithms to handle efficiently.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! How to exploit symmetries to prove that there exist some CSPs near the phase transition cannot be solved within polynomial time?'''},\n",
        "]\n",
        "\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "wqPeXd_FbVM2",
        "outputId": "3ff0137c-b75c-4632-853c-c88e94be679e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"Exploiting symmetries to prove that there exist some CSPs near the phase transition that cannot be solved within polynomial time is a challenging task. However, we can provide a general approach that might help in constructing such instances.\\n\\n1. Study the problem structure: Analyze the structure of the CSP instances near the phase transition. Observe the properties and patterns, such as symmetries, that make these instances harder to solve. \\n\\n2. Construct instances with strong symmetries: Design CSP instances near the phase transition that have a high degree of symmetry, which would make them challenging for algorithms to solve. For example, you could create instances where the constraints are symmetric under certain transformations (e.g., rotations, reflections, or permutations).\\n\\n3. Prove the hardness of these instances: To show that these instances cannot be solved within polynomial time, you can try to reduce a known NP-hard problem to the symmetric CSP instances you've constructed. This would imply that if there were a polynomial-time algorithm for solving these instances, it would also solve the known NP-hard problem, which contradicts the assumption that P ≠ NP.\\n\\n4. Show the existence: Finally, demonstrate that these symmetric CSP instances exist near the phase transition. This could be done through a combination of mathematical analysis and empirical evidence, such as generating random CSP instances near the phase transition and showing that a significant number of them have the desired symmetry properties.\\n\\nIt's important to note that this approach does not guarantee a proof that some CSPs near the phase transition cannot be solved within polynomial time. However, it provides a framework for constructing more complex instances and exploring their hardness, which could potentially lead to new insights and results in the study of phase transitions and computational complexity.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 11\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we prove that there exist instances of the CSP near the phase transition, can we prove that some of them cannot be solved within polynomial time? If not, how to construct more complex instances near the phase transition?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Exploit symmetries and other properties: Some instances of CSPs can be made more difficult by exploiting symmetries or other properties that make them harder to solve. For example, problems with a high degree of symmetry may be difficult for certain algorithms to handle efficiently.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! How to exploit symmetries to prove that there exist some CSPs near the phase transition cannot be solved within polynomial time?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Construct instances with strong symmetries: Design CSP instances near the phase transition that have a high degree of symmetry, which would make them challenging for algorithms to solve. For example, you could create instances where the constraints are symmetric under certain transformations (e.g., rotations, reflections, or permutations).'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! The term 'symmetry' seems to be ambiguous. By 'instances with strong symmetries', do you mean concepts similar to fixed points?'''},\n",
        "]\n",
        "\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "KaaVWer2jQ3k",
        "outputId": "cecb087b-cc1a-4b43-8db4-3b6c28705cba"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "['Apologies for the confusion. By \"instances with strong symmetries,\" I mean instances that exhibit certain structural properties or regularities that make them harder to solve. This could include fixed points, but it is not limited to that concept. \\n\\nSymmetries in CSPs can refer to various types of invariance under transformations. For example, a CSP instance could be symmetric under:\\n\\n1. Permutations: The variables and their assignments can be reordered without affecting the satisfaction of the constraints.\\n2. Rotations or Reflections: The problem can be transformed geometrically (e.g., by rotating or reflecting a graph) without changing the satisfaction of the constraints.\\n3. Duality: A transformation that interchanges the roles of variables and constraints or domain values, while preserving the structure of the problem.\\n\\nThese symmetries can make it more difficult for algorithms to distinguish between different solutions or to efficiently search the solution space. Exploiting such symmetries in CSP instances near the phase transition could help demonstrate the existence of problems that cannot be solved within polynomial time. However, proving that such instances exist and are hard to solve would require further analysis and rigorous mathematical proof.']\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 12\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose now that we have proven the existence of a class of difficult CSP problems that are easy to experience phase transitions (where solvability changes rapidly) and have invariance under transformations, like fixed points. How can we prove that this type of problem cannot be solved in a given time complexity rigorously? Please provide 10 possible ideas.'''},\n",
        "]\n",
        "\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3n3FYlIekbPp",
        "outputId": "93fe27c8-3284-4952-9be1-b2031341eeda"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"1. Reduction to a known hard problem: Show that the given class of difficult CSP problems can be reduced to a known problem that has already been proven to have a certain time complexity. This will imply that the given class of problems cannot be solved faster than the known hard problem.\\n\\n2. Lower bound techniques: Use lower bound techniques like time hierarchy theorems and communication complexity to establish a lower bound on the time complexity of the given class of problems. If the lower bound is higher than the given time complexity, then it is impossible to solve the problem in that time.\\n\\n3. Circuit complexity: Analyze the circuit complexity of the problem. If it can be proven that the problem requires circuits of a certain size, it would imply a lower bound on the time complexity of the problem.\\n\\n4. Fine-grained complexity: Investigate fine-grained complexity assumptions that are based on conjectures about the hardness of specific problems. Use these conjectures to build a reduction from the given class of problems, proving that they cannot be solved within the given time complexity.\\n\\n5. Proof by contradiction: Assume that the given class of problems can be solved within the given time complexity, and then derive a contradiction. This would establish that the problem cannot be solved within the specified time complexity.\\n\\n6. Parameterized complexity: Analyze the problem's parameterized complexity by identifying the parameters that affect the time complexity of the problem. Show that the problem is hard for some specific values of these parameters, implying that the problem cannot be solved in the given time complexity.\\n\\n7. Space-time tradeoffs: Investigate space-time tradeoffs for the given class of problems. If it can be shown that any algorithm solving the problem within the given time complexity must use an infeasible amount of space, this would imply that the problem cannot be solved in the given time complexity.\\n\\n8. Randomized lower bounds: Use randomized lower bounds to show that even randomized algorithms cannot solve the problem within the given time complexity. This will imply that deterministic algorithms also cannot solve the problem in the given time complexity.\\n\\n9. Oracle separation: Assume the existence of an oracle that can solve some other hard problem in polynomial time. Show that even with access to this oracle, the given class of problems cannot be solved within the given time complexity.\\n\\n10. Algebraic complexity: Investigate the algebraic complexity of the problem by analyzing the algebraic computation tree model. Show that any algorithm solving the problem requires a certain number of algebraic operations, implying a lower bound on the time complexity of the problem.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 13\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Suppose now that we have proven the existence of a class of difficult CSP problems that are easy to experience phase transitions (where solvability changes rapidly) and have invariance under transformations, like fixed points. How can we prove that this type of problem cannot be solved in a given time complexity rigorously? Please provide 10 possible ideas.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Proof by contradiction: Assume that the given class of problems can be solved within the given time complexity, and then derive a contradiction. This would establish that the problem cannot be solved within the specified time complexity.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! Please design an idea for proof by contradiction.'''},\n",
        "]\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "tr_VL_-9qoXj",
        "outputId": "210be635-3ef2-4bef-c61b-5149a762b635"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"Idea for Proof by Contradiction:\\n\\n1. Assume that there exists an algorithm A that can solve the given class of difficult CSP problems within the given time complexity T(n), where n is the input size.\\n\\n2. Identify a transformation function F that maps instances of the problem to other instances while preserving the problem's inherent difficulty and invariance under transformation.\\n\\n3. Let P be a difficult CSP problem instance from the class that exhibits phase transitions, and let Q be its transformed version using the function F such that Q = F(P).\\n\\n4. Since our assumption states that algorithm A can solve P within time complexity T(n), it should also be able to solve Q within the same given time complexity T(n), as Q has the same inherent difficulty as P.\\n\\n5. Now, given that P experiences phase transitions, there exists a critical region (a point or tiny interval) where the solvability of P drastically changes. Let's call this critical region C.\\n\\n6. By the invariance property of the transformation function F, there exists a critical region C' for Q such that C' = F(C). Now, in the critical region C', the solvability of Q also changes rapidly.\\n\\n7. Since A can solve P within time complexity T(n), it should be able to solve instances of P in the critical region C within T(n) as well. Similarly, it should be able to solve instances of Q in the critical region C' within T(n).\\n\\n8. However, due to the rapid change in solvability in the critical regions C and C', algorithm A will face difficulty in consistently solving instances of P and Q within the given time complexity T(n). This is because the rapid change in solvability implies that small changes in the input can lead to a dramatic increase in the difficulty of solving the problem.\\n\\n9. This contradicts our initial assumption that algorithm A can solve the given class of difficult CSP problems within the given time complexity T(n).\\n\\n10. Therefore, our initial assumption is false, and there is no algorithm that can solve the given class of difficult CSP problems within the given time complexity T(n) for all instances, including those in the critical regions where solvability changes rapidly.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 14\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''We want to prove that P != NP by contradiction, please list several possible ideas.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''If we could find a specific NP-hard problem and show that it cannot be solved in polynomial time, we could establish P != NP.'''},\n",
        "{\"role\": \"user\", \"content\": '''How to construct such problems?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Identify a well-known NP-complete problem: Start with a known NP-complete problem, such as the Traveling Salesman Problem (TSP), Boolean Satisfiability (SAT), or the Clique problem. These problems have been extensively studied and are known to be hard to solve.\\n\\n2. Reduction: Transform the chosen NP-complete problem into a new problem while preserving its NP-hard properties. This is done through a polynomial-time reduction, which converts an instance of the original problem into an instance of the new problem in such a way that the solutions to both instances correspond to each other.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great idea! What problems can we convert SAT problems to? List as many problems as possible.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''There are many problems to which Boolean Satisfiability (SAT) problems can be reduced. Some of these problems include:\\n\\n1. 3-SAT: A special case of SAT where each clause contains exactly three literals. Despite its restricted nature, 3-SAT is also NP-complete.\\n\\n2. MAX-SAT: Given a Boolean formula in CNF, find the maximum number of clauses that can be satisfied simultaneously by an assignment.\\n\\n3. Graph Coloring: Given an undirected graph and a number k, determine if there is a way to color the vertices with k different colors such that no two adjacent vertices share the same color.\\n\\n4. Clique: Given an undirected graph and a number k, determine if there is a complete subgraph (clique) of size k.\\n\\n5. Hamiltonian Cycle: Given a directed or undirected graph, determine if there is a cycle that visits each vertex exactly once.\\n\\n6. Vertex Cover: Given an undirected graph and a number k, determine if there is a set of k vertices that includes at least one endpoint of each edge.\\n\\n7. Subset Sum: Given a set of integers and a target value, determine if there is a subset of the integers that sums to the target value.\\n\\n8. Knapsack Problem: Given a set of items, each with a weight and a value, determine the most valuable subset of items that can be packed into a knapsack of limited capacity.\\n\\n9. Scheduling problems: Various scheduling problems, such as job shop scheduling and task scheduling in multiprocessor systems, can be reduced from SAT problems.\\n\\n10. Constraint Satisfaction Problems (CSPs): Many CSPs, such as Sudoku, map coloring, and crossword puzzles, can be reduced from SAT problems.\\n\\nThis is not an exhaustive list, but it demonstrates the versatility of SAT problems and their connections to a wide range of other problems.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great idea! What is the time complexity of the process of converting between CSP problems and SAT problems?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''The time complexity of converting between Constraint Satisfaction Problems (CSPs) and SAT problems depends on the specific CSP problem being considered and the encoding technique used. In general, the conversion process can be carried out in polynomial time.\\n\\nHere's an outline of the general process for converting a CSP to a SAT problem:\\n\\n1. Variables: For each variable in the CSP, create a corresponding Boolean variable in the SAT problem. If the CSP variable has a domain size greater than 2, introduce additional Boolean variables to represent each possible value the CSP variable can take.\\n\\n2. Constraints: For each constraint in the CSP, create a corresponding clause in the SAT problem. This may involve converting complex constraints into simpler conjunctions and disjunctions of literals and possibly introducing auxiliary variables.\\n\\nWhen using a proper encoding technique, the number of Boolean variables and clauses introduced will be polynomial in the size of the CSP problem. Therefore, the overall time complexity of converting a CSP to a SAT problem is polynomial.\\n\\nThe reverse process, converting a SAT problem to a CSP, can also be done in polynomial time. The SAT problem's Boolean variables and clauses can be mapped to CSP variables with binary domains and constraints defined by the clauses, respectively.'''},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we construct some extremely difficult CSPs (Constraint Satisfaction Problems), how can we prove that they cannot be solved within polynomial time? Please provide some possible ideas.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Phase Transitions: In some cases, it might be possible to identify phase transitions in the problem instances, where the difficulty of solving the problem rapidly changes from easy to hard as some parameter varies. By studying these transitions, you might be able to show that certain instances of the CSP are inherently difficult and cannot be solved in polynomial time.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! How to prove that some CSPs cannot be solved in polynomial time by studying phase transition?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''To prove that some CSPs cannot be solved in polynomial time by studying phase transitions, you can follow these steps:\\n\\n1. Identify a parameter that affects the problem's difficulty: The first step is to identify a parameter associated with the CSP that influences its difficulty. This parameter could be related to the problem's size, density, or some other structural property. For example, in graph coloring problems, the ratio of edges to nodes may be a relevant parameter.\\n\\n2. Analyze the phase transition: Investigate how the problem's difficulty changes as the parameter varies. Observe at which point the problem transitions from being easy to solve to becoming hard. This is the phase transition point. Typically, phase transitions in CSPs exhibit a sudden change in the solvability of problem instances.\\n\\n3. Show the existence of hard instances: Prove that there exist instances of the CSP near the phase transition point that are difficult to solve. You can do this by showing that the problem instances exhibit certain properties or structures that make them inherently hard. This might involve relating the CSP to a well-known hard problem, such as an NP-complete problem, or establishing the presence of a lower bound on the complexity of the instances.\\n\\n4. Relate the difficulty to polynomial time solvability: With the existence of hard instances established, you must now relate this difficulty to the inability to solve the problem in polynomial time. One way to achieve this is by showing that any algorithm capable of solving the hard instances in polynomial time would also be able to solve other known hard problems in polynomial time, which is unlikely or considered impossible under the widely believed conjecture P ≠ NP.\\n\\nKeep in mind that this method does not provide a formal proof that the problem is not solvable in polynomial time, but it can provide strong evidence to support the intractability of the CSP under consideration. To obtain a more rigorous proof, you would need to establish a direct connection between your CSP and a known hard problem, such as by showing a reduction from an NP-complete problem to your CSP.'''},\n",
        "{\"role\": \"user\", \"content\": '''Suppose we prove that there exist instances of the CSP near the phase transition, can we prove that some of them cannot be solved within polynomial time? If not, how to construct more complex instances near the phase transition?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Exploit symmetries and other properties: Some instances of CSPs can be made more difficult by exploiting symmetries or other properties that make them harder to solve. For example, problems with a high degree of symmetry may be difficult for certain algorithms to handle efficiently.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! How to exploit symmetries to prove that there exist some CSPs near the phase transition cannot be solved within polynomial time?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Construct instances with strong symmetries: Design CSP instances near the phase transition that have a high degree of symmetry, which would make them challenging for algorithms to solve. For example, you could create instances where the constraints are symmetric under certain transformations (e.g., rotations, reflections, or permutations).'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! The term 'symmetry' seems to be ambiguous. By 'instances with strong symmetries', do you mean concepts similar to fixed points?'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Apologies for the confusion. By \"instances with strong symmetries,\" I mean instances that exhibit certain structural properties or regularities that make them harder to solve. This could include fixed points, but it is not limited to that concept. \\n\\nSymmetries in CSPs can refer to various types of invariance under transformations. For example, a CSP instance could be symmetric under:\\n\\n1. Permutations: The variables and their assignments can be reordered without affecting the satisfaction of the constraints.\\n2. Rotations or Reflections: The problem can be transformed geometrically (e.g., by rotating or reflecting a graph) without changing the satisfaction of the constraints.\\n3. Duality: A transformation that interchanges the roles of variables and constraints or domain values, while preserving the structure of the problem.\\n\\nThese symmetries can make it more difficult for algorithms to distinguish between different solutions or to efficiently search the solution space. Exploiting such symmetries in CSP instances near the phase transition could help demonstrate the existence of problems that cannot be solved within polynomial time. However, proving that such instances exist and are hard to solve would require further analysis and rigorous mathematical proof.'''},\n",
        "{\"role\": \"user\", \"content\": '''Suppose now that we have proven the existence of a class of difficult CSP problems that are easy to experience phase transitions (where solvability changes rapidly) and have invariance under transformations, like fixed points. How can we prove that this type of problem cannot be solved in a given time complexity rigorously? Please provide 10 possible ideas.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Proof by contradiction: Assume that the given class of problems can be solved within the given time complexity, and then derive a contradiction. This would establish that the problem cannot be solved within the specified time complexity.'''},\n",
        "{\"role\": \"user\", \"content\": '''Great! Please design an idea for proof by contradiction.'''},\n",
        "{\"role\": \"assistant\", \"content\": '''Idea for Proof by Contradiction:\\n\\n1. Assume that there exists an algorithm A that can solve the given class of difficult CSP problems within the given time complexity T(n), where n is the input size.\\n\\n2. Identify a transformation function F that maps instances of the problem to other instances while preserving the problem's inherent difficulty and invariance under transformation.\\n\\n3. Let P be a difficult CSP problem instance from the class that exhibits phase transitions, and let Q be its transformed version using the function F such that Q = F(P).\\n\\n4. Since our assumption states that algorithm A can solve P within time complexity T(n), it should also be able to solve Q within the same given time complexity T(n), as Q has the same inherent difficulty as P.\\n\\n5. Now, given that P experiences phase transitions, there exists a critical region (a point or tiny interval) where the solvability of P drastically changes. Let's call this critical region C.\\n\\n6. By the invariance property of the transformation function F, there exists a critical region C' for Q such that C' = F(C). Now, in the critical region C', the solvability of Q also changes rapidly.\\n\\n7. Since A can solve P within time complexity T(n), it should be able to solve instances of P in the critical region C within T(n) as well. Similarly, it should be able to solve instances of Q in the critical region C' within T(n).\\n\\n8. However, due to the rapid change in solvability in the critical regions C and C', algorithm A will face difficulty in consistently solving instances of P and Q within the given time complexity T(n). This is because the rapid change in solvability implies that small changes in the input can lead to a dramatic increase in the difficulty of solving the problem.\\n\\n9. This contradicts our initial assumption that algorithm A can solve the given class of difficult CSP problems within the given time complexity T(n).\\n\\n10. Therefore, our initial assumption is false, and there is no algorithm that can solve the given class of difficult CSP problems within the given time complexity T(n) for all instances, including those in the critical regions where solvability changes rapidly.'''},\n",
        "{\"role\": \"user\", \"content\": '''Please now organize all our historical conversations and sort out a general proof idea.'''},\n",
        "]\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n",
        "\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "ezT3dKqfyE0w",
        "outputId": "37df8759-7ea5-436d-cd0f-a5e0d96607c9"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[\"Here's a general proof idea based on our historical conversations:\\n\\n1. Identify a well-known NP-complete problem, such as SAT, which has been extensively studied and known to be hard to solve.\\n\\n2. Convert the NP-complete problem into a CSP problem using a polynomial-time reduction that preserves the problem's inherent difficulty.\\n\\n3. Study the phase transition behavior of the resulting CSP problem by identifying a parameter associated with the problem that influences its difficulty. Investigate how the problem's difficulty changes as the parameter varies and identify the phase transition point where the problem transitions from being easy to hard.\\n\\n4. Construct instances of the CSP problem near the phase transition point that exhibit strong symmetries or invariance under specific transformations, such as fixed points, permutations, rotations, reflections, or duality.\\n\\n5. Prove the existence of hard instances of the CSP problem near the phase transition point by showing that certain instances exhibit properties or structures that make them inherently difficult to solve. This might involve relating the CSP to a well-known hard problem or establishing the presence of a lower bound on the complexity of the instances.\\n\\n6. Design a proof by contradiction to show that the hard instances of the CSP problem near the phase transition point cannot be solved within a given time complexity. Assume that there exists an algorithm that can solve the problem within the given time complexity, and derive a contradiction by considering the phase transition behavior, symmetries, and invariance properties of the problem instances.\\n\\n7. Conclude that the hard instances of the CSP problem near the phase transition point cannot be solved within the given time complexity. This provides evidence supporting the intractability of the CSP problem and the claim that P ≠ NP.\\n\\nKeep in mind that this general proof idea provides a framework for exploring the complexity of CSP problems and their relationship to NP-complete problems. A rigorous proof of the intractability of specific CSP problems would require a more extensive analysis and mathematical proof.\"]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# 15\n",
        "messages=[\n",
        "{\"role\": \"system\", \"content\": \"You are a mathematician and computer theory expert, good at innovation and thinking.\"},\n",
        "{\"role\": \"user\", \"content\": '''Can you list some models that can generate hard examples for Constraint Satisfaction Problems (CSP)?'''},\n",
        "]\n",
        "\n",
        "\n",
        "outputs, messages = gpt4_with_history(messages, max_tokens=1024)\n",
        "print(outputs)\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2uYezTvH3ojR",
        "outputId": "d67fb124-7c93-49d9-eeae-e27c537a66d5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "['Constraint Satisfaction Problems (CSP) involve finding values for a set of variables subject to a set of constraints. Some models and techniques are designed to generate hard instances of CSPs, making it challenging for algorithms to solve them efficiently. These hard instances are useful for evaluating and comparing the performance of various algorithms.\\n\\n1. Phase Transition Model: CSPs exhibit a phase transition phenomenon where the problem goes from being mostly easy to mostly hard as the density of constraints increases. The phase transition model generates hard instances by tuning the constraint density to be around the phase transition region. For example, in the case of random k-SAT problems, this occurs when the ratio of constraints to variables (clause-to-variable ratio) is around a critical value.\\n\\n2. Model RB: Model RB is a random binary CSP generator that focuses on the backbone size, the number of variables that must be assigned a specific value in any solution. By controlling the backbone size, Model RB can generate hard instances with a small backbone size, which makes them difficult to solve.\\n\\n3. Hidden Structure Model: This model generates hard instances by introducing hidden structure in CSPs. For example, problems with an underlying graph structure (e.g., graph coloring) can be made more challenging by hiding clique-like substructures within the graph. The hidden structure makes it difficult for algorithms to exploit the natural structure of the problem.\\n\\n4. Quasigroup Completion Problem (QCP): QCP is an example of a specific CSP that can generate hard instances. In this problem, a partially filled Latin square must be completed by assigning the missing values while preserving the Latin square property (i.e., each value appears exactly once in each row and column).\\n\\n5. Magic Square Problem: Another specific CSP that can produce hard instances is the magic square problem, where the task is to fill a square grid with distinct integers such that the sum of the numbers in each row, column, and diagonal is the same. The problem becomes harder as the size of the square increases.\\n\\nThese models and problems can generate hard instances for CSPs that are particularly useful for benchmarking and testing the performance of constraint solvers and heuristic algorithms.']\n"
          ]
        }
      ]
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "name": "python3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}